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It is well-known that spectral singularities play a certain critical role in the spectral theory of nonselfadjoint differential operators. Their existence is accompanied by specific phenomenon which are new in the sense that they do not occur either in the spectral theory of self-adjoint or normal operators. So, in this paper, we examine spectrum and spectral singularities of the impulsive quadratic pencil of Schrödinger operators. In particular, we study the dependence of the structure of discrete spectrum and spectral singularities of these operators of the behaviour of the potential functions at infinity. We present some conditions on potential functions which guarantee that the impulsive quadratic pencil of Schrödinger operators have a finite number of eigenvalues and spectral singularities with finite multiplicities. Later, we discuss the properties of principal functions corresponding to the spectral singularities. [ABSTRACT FROM AUTHOR]